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A366227
O.g.f. A(x) satisfies: A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.
0
1, 1, 6, 138, 8648, 1272948, 424058592, 334836466656, 728593565874816, 5632989888855720864, 184539760855097635059200, 25027477244647424010315231744, 13206715998089387470949589465286656, 26431031766456352400292737393044784872448, 199091399877503863934385670788355318673030504448
OFFSET
0,3
COMMENTS
In general, we have the following identity:
given A(x) = Sum_{n>=0} a(n)*x^n satisfies
A(x) = 1 + x*Sum_{n>=0} p^n * log( A(q^n*x) )^n / n!,
then a(n+1) = [x^n] A(x)^(p*q^n) for n >= 0, with a(0)=1,
for arbitrary fixed parameters p and q.
Here, p = 3 and q = 2.
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*Sum_{n>=0} 3^n * log( A(2^n*x) )^n / n!.
(2) a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0, with a(0)=1.
EXAMPLE
G.f.: A(x) = 1 + x + 6*x^2 + 138*x^3 + 8648*x^4 + 1272948*x^5 + 424058592*x^6 + 334836466656*x^7 + 728593565874816*x^8 + ...
where
A(x) = 1 + x*[1 + 3*log(A(2*x)) + 3^2*log(A(2^2*x))^2/2! + 3^3*log(A(2^2*x))^3/3! + ... + 3^n*log(A(2^n*x))^n/n! + ...].
RELATED SERIES.
log(A(x)) = x + 11*x^2/2 + 397*x^3/3 + 33991*x^4/4 + 6318201*x^5/5 + 2536406543*x^6/6 + 2340834765809*x^7/7 + ...
RELATED TABLE.
The table of coefficients of x^k in A(x)^(3*2^n) begins:
n=0: [1, 3, 21, 451, 26898, 3876222, ...];
n=1: [1, 6, 51, 1028, 56943, 7932774, ...];
n=2: [1, 12, 138, 2668, 128823, 16653720, ...];
n=3: [1, 24, 420, 8648, 340722, 37135560, ...];
n=4: [1, 48, 1416, 37456, 1272948, 97890096, ...];
n=5: [1, 96, 5136, 210848, 8146728, 424058592, ...];
n=6: [1, 192, 19488, 1407808, 83154768, 4578119616, 334836466656, ...]; ...
in which the main diagonal equals this sequence shift left,
illustrating that a(n+1) = [x^n] A(x)^(3*2^n) for n >= 0.
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, Vec(Ser(A)^(3*2^(#A-1)))[ #A])); A[n+1]}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + x*sum(m=0, #A, 3^m*log( subst(Ser(A), x, 2^m*x +x*O(x^n)))^m/m!) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 17 2023
STATUS
approved